I'm not even using coc.nvim (which works with a derivative of the Code extension)

Running requires custom client subcommand, so we probably don't

I'm not sure if I understood the question, but I'm able to run tests from vim via CodeLens actions. I'm using LanguageClient-neovim which gets the rust-analyzer.runSingle command and runs the test in a terminal. Same goes for running the binary itself.